Understanding self-focusing of lasers using amplitude equations
07 / 2002 - unknown
The qualitative study of nonlinear dynamical systems is very important in understanding many problems in science and technology. We focus on projects that are motivated by concrete applications in laser dynamics. Lasers are highly nonlinear devices and the competition between nonlinear and dispersive effects leads to very interesting dnamics. In particular, a phenomenon known as self-focusing can occur. Self-focusing, also known as wave collapse or blowup, is characterised by a spatial contraction taking place together with an extreme increase of the field amplitude. It arises when nonlinear effects dominate the dispersive ones, up to the formation of the singularity. The central theme in this proposal is the observation that the phenomena of dynamics in semiconductor lasers related to self-focusing and external influences can be described using amplitude equations. In general, amplitude equations are nonlinear partial differential equations for complex-valued functions that describe the evolution of the amplitude of a wave close to an instability formation or bifurcation. We will develop mathematical methods for studying amplitude equations with an emphasis on blowup structures. In addition, we believe that our approach can also be applied to the analysis of the dynanmics of lasers where delayed optical feedback is present as an external influence. We will use techniques from the field of dynamical systems combined with numerical methods.